To illustrate the effects of bubbles on heat and mass transport, comparison will be made to a simple base case in which all gases are assumed to escape from the thermally degrading sample instantly upon gasification. The local volume fractions of polymer and gas under this assumption are constant over time and space with values φp = 1 and φg = 0 respectively. A potential singularity arises from the instantaneous transport of gas to the surface, making gas radial velocity Wg infinite. This difficulty is resolved by the observation from the mass balance equations (2) and (3) that the product φgWg is finite. For polymer and gas densities ρp and ρg constant, this product is
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(68) |
In this case, therefore, the barycentric velocity W from equation (6) becomes
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(69) |
and the energy equation (7) becomes
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(70) |
This energy equation may appear to neglect the transport of heat due to the inward movement of the molten polymeric material to fill in the space left by the escaping gas, as has been argued by Staggs [28] for a rectangular geometry. As the analysis makes clear, however, the inward transport of heat energy by the polymer melt is balanced by the outward transport of heat by the escaping gases. Although gases escape instantaneously in this simple model, heat transport due to gas convection cannot be neglected in the equation of energy conservation.
The radius of a spherical PMMA sample exposed to a uniform external heat flux of 60 W/cm2 is shown as a function of time in Figure 9, with the corresponding mass loss rate vs. time in Figure 10. Initially upon exposure to heat, the sample undergoes preheating. During this time, roughly the first eight seconds in this case, the sample radius remains at the initial value and the mass loss rate is zero. Heat conduction raises internal temperatures, as shown in Figure 11, until they are high enough to cause decomposition of the polymer. The temperature profile within the sample at two second intervals is plotted in Figure 12.
By about ten seconds, as indicated by the knee in the plot of center temperature vs. time in Figure 11, the entire sphere is hot enough to undergo gasification, and the pyrolysis process enters an extended period during which radius decreases linearly with time, surface temperature is nearly steady, and center temperature rises at a slower rate than that during preheating. During this time, the heat energy entering the spherical sample is divided between gasification and conduction.
During the final few seconds of pyrolysis, the temperature becomes uniform throughout, and the energy entering the spherical sample is expended in raising its temperature as well as in gasifying the small amount of polymer remaining.
Figure 9: Sample radius vs. time for a spherical PMMA sample exposed to an external heat flux of 60 W/cm2, assuming instantaneous escape of gases during degradation.
Figure 10: Mass loss rate vs. time corresponding to Figure 9.
Figure 11: Temperature vs. time at the center (lower line) and on the outer surface of the spherical sample.
Figure 12: Temperature vs. radius from time t = 0 (straight line at T = 298 K) to completion of pyrolysis at intervals of 2 s.
